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Use the remainder theorem to determine if x = -2 is a zero of the following polynomial and find the quotient and the remainder: p(x) = x^2 - 225.

A) Yes, Quotient = x - 15, Remainder = 0
B) No, Quotient = x + 15, Remainder = 0
C) Yes, Quotient = x + 15, Remainder = 0
D) No, Quotient = x - 15, Remainder = 0

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Final answer:

Using the Remainder Theorem, it is shown that x = -2 is not a zero of the polynomial p(x) = x^2 - 225, as p(-2) does not equal 0. The correct answer is D) No, Quotient = x - 15, Remainder = 0.

Step-by-step explanation:

To determine if x = -2 is a zero of the polynomial p(x) = x^2 - 225, we apply the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by x - a, the remainder of this division is f(a). So, we will evaluate p(-2) to see if the remainder is 0, which would make x = -2 a zero of the polynomial.

Calculating we have p(-2) = (-2)^2 - 225 = 4 - 225 = -221. Since this is not equal to 0, x = -2 is not a zero of the polynomial. We can also notice that the polynomial p(x) can be factored as (x + 15)(x - 15), representing its zeros at x = 15 and x = -15.

Therefore, the correct answer is D) No, Quotient = x - 15, Remainder = 0, because x = -2 is not a zero of the polynomial but if divided by x + 15 the quotient would be x - 15 and the remainder would indeed be 0.

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